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This study describes a novel application of max-plus linear algebra to the timing of digital hardware. The authors give a rigorous, algorithmic approach to `time borrowing' - a technique whereby the use of a multi-phase clock can allow for a more flexible, efficient use of time. In this approach the system is clocked periodically, but within each clock cycle processes interact asynchronously, allowing longer processes to be juxtaposed with shorter processes. The authors have shown that this problem can be solved completely using linear algebra defined over the max-plus semi-ring. This work establishes the mathematical foundation of an earlier, heuristic approach to the problem.